For example, as far as I'm concerned Admiral hanson is Anika Hanson's (seven of nine)'s grandfather. makes his push for defeating the Borg all the more poignant. What are some of yours?
Star Trek V was all a dream. The Intrepid class has a shuttle manufacturing plant onboard. Kira's and T'Pols uniforms only look like ridiculous catsuits on camera. Seven wears normal 24th century business attire. Picard and Beverly hooked up at some point after Attached, and just kept in on the DL. Roddeberry himself said much of TOS was apocryphal, so I tend to pick over episodes in my head saying "this one's canon" or "this one's a fun story that couldn't possibly have happened."
Ther Klingon Empire is really just like the novel Final Reflection. The Romulan Empire is really just like the novel My Enemy My Ally. Federation history is really just like the old FASA role playing game.
All problematic ENT episodes are Riker playing on the holodeck. All problematic TNG and VOY episodes are Barclay playing on the holodeck. All problematic TOS or DS9 episodes are Bashir, O'Brien, and/or Garak playing in the holosuite. And Star Trek V was just a dream.
Mine: During the TOS era, Starfleet adopted a policy of using painted, standardized parts (saucer sections, engineering sections, and nacelles) for its starships and the original Constitution-class was a product of that time. Robert April was the first captain of the USS Enterprise (NCC-1701); Lieutenant Commander Christopher Pike was his first officer. Fashion in the mid 23rd-Century underwent a retro period inspired by the 1960s. Phase weaponry never entered widespread usage in the late 22nd-Century and Starfleet relied on a new form of "laser" weaponry until the 2260s. The USS Constellation (NCC-1017) originally came from a different starship class but was ultimately upgraded to a Constitution-class. Kirk commanded a second five-year mission after TMP. Following the completion of Kirk's second 5-year mission, he was forced to return to the admiralty as the Commandant of Starfleet Academy; Spock was subsequently promoted to Captain of the Enterprise. At some point prior to TWOK, the Enterprise was assigned to Starfleet Academy as a cadet training vessel still under Spock's command. Relations between the Federation and the Klingons took a sharp turn for the worse after Star Trek V and soured the views of many Starfleet veterans (including Kirk and Scotty) towards Klingons. The Enterprise-B was in the final months of her construction at the time of Star Trek VI. John Harriman was only the first captain of the Enterprise-B, there were several others that came after him. The Enterprise-B was decommissioned after 50 years as the longest-serving Federation starship to bear the Enterprise name. Riker and LaForge had attended a class together as cadets at Starfleet Academy, but didn't really get to know each other during their time there. The first seven years of the Enterprise-D's service was spent performing a wide variety of missions. The ship was about to finally embark on a long-term (10+ year) deep-space exploration mission at the time of Generations.
Fandom Star Trek Chronology, Version 17. Each Trek spinoff series exists in a Slightly different timeline... In TOS, the Eugenics Wars in the 1990s was WW III... In TNG, WW III was in the mid-21st Century... DS9 had Section 31, etc... Enterprise was a damaged timeline and/or a pseudo-historical holonovel viewed by Deanna and Will Riker... Star Trek (2009) Never Happened... Ditto for Into Darkness, which can only be worse... Building on the above, maybe holo-adventures for Deanna's and Bev's kids to play in...
Most of these gel pretty well with what I've always imagined (except for a second 5-year-mission. I've never seen the need for it.) Also, in my continuity, women are wearing miniskirts again.
Star Trek, and all of its incarnations, are simply stories made to entertain the masses. Some like a little, others like a little more. That's my favorite way to deal with the inconsistencies.